On the -Biharmonic Operator with Critical Sobolev Exponent and Nonlinear Steklov Boundary Condition

We show that this operator possesses at least one nondecreasing sequence of positive eigenvalues. A direct characterization of the principal eigenvalue (the first one) is given that we apply to study the spectrum of the -biharmonic operator with a critical Sobolev exponent and the nonlinear Steklov boundary conditions using variational arguments and trace critical Sobolev embedding. 1. Introduction Let be a smooth bounded domain in . Consider the fourth-order nonlinear Steklov boundary eigenvalue problem where is the critical Sobolev exponent defined by Here is a real parameter which plays the role of an eigenvalue. is the operator of fourth order called the -biharmonic operator. For , the linear operator . is the iterated Laplacian that multiplied with positive constant appears often in Navier-Stokes equations as being a viscosity coefficient. Its reciprocal operator, denoted by , is celebrated Green’s operator [1]. The nonlinear boundary condition describes a nonlinear flux through the boundary which depends on the solution itself and its normal derivation. Here denotes the outer normal derivative of on defined by . Notice that the biharmonic equation , corresponding to , is a partial differential equation of fourth order which appears in quantum mechanics and in the theory of linear elasticity modeling Stokes’ flows. It is well known that elliptic problems with eigenvalues in the boundary conditions are usually called Steklov problems from their first appearance in [2]. For the fourth-order Steklov eigenvalue problems, the first eigenvalue plays a crucial role in the positivity preserving property for the biharmonic operator under conditions , on (see [3]). In [4], the authors investigated the bound for the first eigenvalue on the plane square and proved that the first eigenvalue is simple and its eigenfunction does not change sign. The authors of [5, 6] studied the spectrum of a fourth-order Steklov eigenvalue problem on a bounded domain in and gave the explicit form of the spectrum in the case where the domain is a ball. Let us mention that the spectrum of the fourth order Steklove has been completely determined by Ren and Yang [7] in the case , using the theory of completely continuous operators. It is already evident from the well-studied second-order case that nonlinear equations with critical growth terms present highly interesting phenomena concerning the existence and nonexistence. For the fourth-order equations is more challenging, since the techniques depend strongly on the imposed boundary conditions. For the case of , , with , the problem